Possible Derivation of Energy Gaps or Hadron Masses from Fictitious-Time Correlations in Stochastic Quantization Method

Abstract

The Parisi-Wu stochastic quantization method offers us one of the powerful tools for nonperturbative approaches to quantum field theory, especially, for numerical simulations of long-range correlations to give energy gaps or hadron masses. I) The method is so designed as to produce quantum theory as a thermal equilibrium limit of a hypothetical stochastic process with respect to a fictitious time other than the ordinary time. Usually we calculate physical quantities after erasing fictitious time dependences. In fact, in many papers of numerical simulations by means of the stochastic quantization, energy gaps or hadron masses have so far been derived from ordinary space-time correlations but not from fictitious-time ones.l),2) However, the stationary correlation function given by the stochastic quantization method must be subject to a dispersion formula, implicitly existing inside its Fourier transform, which gives a certain relation between the ordinary momentum variables and the fictitious-time frequency. So it would be naturally expected that we can extract information about energy gaps or hadron masses from the fictitioustime dependence of the correlation function. In this paper we discuss such a problem in the case of free field, with some perspective to interacting fields, and then carry out numerical simulations of fictitious-time correlations and energy gaps in the case of a nonrelativistic particle in a fixed potential. Consider a neutral scalar field ¢(x, t) with particle mass m, depending on D-dimensional Euclidean space-time coordinate x and fictitious time t. In the case of free field, stochastic quantization of ¢ (x, t) starts with the Langevin equation